Inthispaper,necessaryandsufficientconditionsforaclosedrangecompositionoperatorC_φonthegeneralfamilyofholomorphicfunctionspacesF(p,q,s)andmoregenerallyonα-BesovtypespacesF(p,αp-2,s)aregiven.WegiveaCarlesonmeasurecharacterizationonF(p,αp-2,s)spaces,thenweindicatehowCarlesonmeasurescanbeusedtocharacterizeboundednessandcompactnessofC_φonF(p,q,s)andF(p,αp-2,s)spaces.
